coding theory - Binary codes with large distance

No. If we take ${-1/sqrt n,1/sqrt n}^n$ instead of ${0,1}^n$, the problem reduces to asking if we can have many unit vectors $v_j$ with pairwise scalar products $-gamma$ or less where $gamma>0$ is a fixed number. But if we have $N$ such vectors, then the square of the norm of their sum is at most $N-gamma N(N-1)$. Since this square must be non-negative, we get $N-1legamma^{-1}$ regardless of the dimension.